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By the Pontryagin–Thom theorem, there is a ring spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109301.png" /> (cf. [[Spectrum of a ring|Spectrum of a ring]]) whose [[Homotopy|homotopy]] is isomorphic to the graded ring of bordism classes of closed smooth manifolds with a complex structure on their stable normal bundles (cf. also [[Cobordism|Cobordism]]). E.H. Brown and F.P. Peterson [[#References|[a1]]] showed that, when localized at a prime <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109302.png" />, the spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109303.png" /> is homotopy equivalent to the wedge of various suspensions (cf. also [[Suspension|Suspension]]) of a ring spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109304.png" />, the Brown–Peterson spectrum. The homotopy of this spectrum is the polynomial algebra
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109305.png" /></td> </tr></table>
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where the degree of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109306.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109307.png" />. As a module over the [[Steenrod algebra|Steenrod algebra]],
+
By the Pontryagin–Thom theorem, there is a ring spectrum  $  MU $(
 +
cf. [[Spectrum of a ring|Spectrum of a ring]]) whose [[Homotopy|homotopy]] is isomorphic to the graded ring of bordism classes of closed smooth manifolds with a complex structure on their stable normal bundles (cf. also [[Cobordism|Cobordism]]). E.H. Brown and F.P. Peterson [[#References|[a1]]] showed that, when localized at a prime  $  p $,
 +
the spectrum  $  MU $
 +
is homotopy equivalent to the wedge of various suspensions (cf. also [[Suspension|Suspension]]) of a ring spectrum  $  BP $,  
 +
the Brown–Peterson spectrum. The homotopy of this spectrum is the polynomial algebra
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109308.png" /></td> </tr></table>
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$$
 +
\pi _ {*} BP = \mathbf Z _ {( p ) }  [ v _ {1} \dots v _ {n} , \dots ] ,
 +
$$
  
Four properties of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b1109309.png" /> have made it one of the most useful spectra in homotopy theory. First, D. Quillen [[#References|[a5]]] determined the structure of its ring of operations. Second, A. Liulevicius [[#References|[a3]]] and M. Hazewinkel [[#References|[a2]]] constructed polynomial generators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093010.png" /> with good properties. Third, the Baas–Sullivan construction can be used to construct simple spectra from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093011.png" /> with very nice properties. The most notable of these spectra are the Morava <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093013.png" />-theories <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093014.png" />, which are central in the statement of the periodicity theorem. (See [[#References|[a7]]] for an account of the nilpotence and periodicity theorems.) Fourth, S.P. Novikov [[#References|[a4]]] constructed the Adams–Novikov [[Spectral sequence|spectral sequence]], which uses knowledge of the Brown–Peterson homology of a spectrum <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093015.png" /> to compute the homotopy of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093016.png" />. (See [[#References|[a6]]] for a survey of how the Adams–Novikov spectral sequence gives information on the stable homotopy groups of spheres.)
+
where the degree of $  v _ {n} $
 +
is  $  2 ( p  ^ {n} - 1 ) $.  
 +
As a module over the [[Steenrod algebra|Steenrod algebra]],
  
An introduction to the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093017.png" /> is given in [[#References|[a8]]].
+
$$
 +
H _ {*} ( BP; \mathbf Z/p ) \simeq \left \{
 +
\begin{array}{l}
 +
{\mathbf Z/2 [ \xi _ {1}  ^ {2} \dots \xi _ {n}  ^ {2} , \dots ] , \  p = 2, } \\
 +
{\mathbf Z/p [ \xi _ {1} \dots \xi _ {n} , \dots ] , \  p  \textrm{ odd  } . }
 +
\end{array}
 +
\right .
 +
$$
 +
 
 +
Four properties of  $  BP $
 +
have made it one of the most useful spectra in homotopy theory. First, D. Quillen [[#References|[a5]]] determined the structure of its ring of operations. Second, A. Liulevicius [[#References|[a3]]] and M. Hazewinkel [[#References|[a2]]] constructed polynomial generators of  $  \pi _ {*} BP $
 +
with good properties. Third, the Baas–Sullivan construction can be used to construct simple spectra from  $  BP $
 +
with very nice properties. The most notable of these spectra are the Morava  $  K $-
 +
theories  $  K ( n ) $,
 +
which are central in the statement of the periodicity theorem. (See [[#References|[a7]]] for an account of the nilpotence and periodicity theorems.) Fourth, S.P. Novikov [[#References|[a4]]] constructed the Adams–Novikov [[Spectral sequence|spectral sequence]], which uses knowledge of the Brown–Peterson homology of a spectrum  $  X $
 +
to compute the homotopy of  $  X $.
 +
(See [[#References|[a6]]] for a survey of how the Adams–Novikov spectral sequence gives information on the stable homotopy groups of spheres.)
 +
 
 +
An introduction to the study of $  BP $
 +
is given in [[#References|[a8]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Brown,  F.P. Peterson,  "A spectrum whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093018.png" />-homology is the algebra of reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093019.png" />th powers"  ''Topology'' , '''5'''  (1966)  pp. 149–154</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hazewinkel,  "Constructing formal groups III. Applications to complex cobordism and Brown–Peterson cohomology"  ''J. Pure Appl. Algebra'' , '''10'''  (1977/78)  pp. 1–18</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Liulevicius,  "On the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093020.png" />" , ''Lecture Notes in Mathematics'' , '''249''' , Springer  (1971)  pp. 47–53</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the viewpoint of cobordism theories"  ''Math. USSR Izv.''  (1967)  pp. 827–913  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Quillen,  "On the formal group laws of unoriented and complex cobordism theory"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 1293–1298</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.C. Ravenel,  "Complex cobordism and stable homotopy groups of spheres" , ''Pure and Applied Mathematics'' , '''121''' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.C. Ravenel,  "Nilpotence and periodicity in stable homotopy theory" , ''Annals of Math. Stud.'' , '''128''' , Princeton Univ. Press  (1992)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W.S. Wilson,  "Brown–Peterson homology, an introduction and sampler" , ''Regional Conf. Ser. Math.'' , '''48''' , Amer. Math. Soc.  (1982)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  E.H. Brown,  F.P. Peterson,  "A spectrum whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093018.png" />-homology is the algebra of reduced <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093019.png" />th powers"  ''Topology'' , '''5'''  (1966)  pp. 149–154</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  M. Hazewinkel,  "Constructing formal groups III. Applications to complex cobordism and Brown–Peterson cohomology"  ''J. Pure Appl. Algebra'' , '''10'''  (1977/78)  pp. 1–18</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A. Liulevicius,  "On the algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110930/b11093020.png" />" , ''Lecture Notes in Mathematics'' , '''249''' , Springer  (1971)  pp. 47–53</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.P. Novikov,  "The methods of algebraic topology from the viewpoint of cobordism theories"  ''Math. USSR Izv.''  (1967)  pp. 827–913  ''Izv. Akad. Nauk SSSR Ser. Mat.'' , '''31'''  (1967)  pp. 855–951</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D. Quillen,  "On the formal group laws of unoriented and complex cobordism theory"  ''Bull. Amer. Math. Soc.'' , '''75'''  (1969)  pp. 1293–1298</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  D.C. Ravenel,  "Complex cobordism and stable homotopy groups of spheres" , ''Pure and Applied Mathematics'' , '''121''' , Acad. Press  (1986)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D.C. Ravenel,  "Nilpotence and periodicity in stable homotopy theory" , ''Annals of Math. Stud.'' , '''128''' , Princeton Univ. Press  (1992)</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  W.S. Wilson,  "Brown–Peterson homology, an introduction and sampler" , ''Regional Conf. Ser. Math.'' , '''48''' , Amer. Math. Soc.  (1982)</TD></TR></table>

Revision as of 06:29, 30 May 2020


By the Pontryagin–Thom theorem, there is a ring spectrum $ MU $( cf. Spectrum of a ring) whose homotopy is isomorphic to the graded ring of bordism classes of closed smooth manifolds with a complex structure on their stable normal bundles (cf. also Cobordism). E.H. Brown and F.P. Peterson [a1] showed that, when localized at a prime $ p $, the spectrum $ MU $ is homotopy equivalent to the wedge of various suspensions (cf. also Suspension) of a ring spectrum $ BP $, the Brown–Peterson spectrum. The homotopy of this spectrum is the polynomial algebra

$$ \pi _ {*} BP = \mathbf Z _ {( p ) } [ v _ {1} \dots v _ {n} , \dots ] , $$

where the degree of $ v _ {n} $ is $ 2 ( p ^ {n} - 1 ) $. As a module over the Steenrod algebra,

$$ H _ {*} ( BP; \mathbf Z/p ) \simeq \left \{ \begin{array}{l} {\mathbf Z/2 [ \xi _ {1} ^ {2} \dots \xi _ {n} ^ {2} , \dots ] , \ p = 2, } \\ {\mathbf Z/p [ \xi _ {1} \dots \xi _ {n} , \dots ] , \ p \textrm{ odd } . } \end{array} \right . $$

Four properties of $ BP $ have made it one of the most useful spectra in homotopy theory. First, D. Quillen [a5] determined the structure of its ring of operations. Second, A. Liulevicius [a3] and M. Hazewinkel [a2] constructed polynomial generators of $ \pi _ {*} BP $ with good properties. Third, the Baas–Sullivan construction can be used to construct simple spectra from $ BP $ with very nice properties. The most notable of these spectra are the Morava $ K $- theories $ K ( n ) $, which are central in the statement of the periodicity theorem. (See [a7] for an account of the nilpotence and periodicity theorems.) Fourth, S.P. Novikov [a4] constructed the Adams–Novikov spectral sequence, which uses knowledge of the Brown–Peterson homology of a spectrum $ X $ to compute the homotopy of $ X $. (See [a6] for a survey of how the Adams–Novikov spectral sequence gives information on the stable homotopy groups of spheres.)

An introduction to the study of $ BP $ is given in [a8].

References

[a1] E.H. Brown, F.P. Peterson, "A spectrum whose -homology is the algebra of reduced th powers" Topology , 5 (1966) pp. 149–154
[a2] M. Hazewinkel, "Constructing formal groups III. Applications to complex cobordism and Brown–Peterson cohomology" J. Pure Appl. Algebra , 10 (1977/78) pp. 1–18
[a3] A. Liulevicius, "On the algebra " , Lecture Notes in Mathematics , 249 , Springer (1971) pp. 47–53
[a4] S.P. Novikov, "The methods of algebraic topology from the viewpoint of cobordism theories" Math. USSR Izv. (1967) pp. 827–913 Izv. Akad. Nauk SSSR Ser. Mat. , 31 (1967) pp. 855–951
[a5] D. Quillen, "On the formal group laws of unoriented and complex cobordism theory" Bull. Amer. Math. Soc. , 75 (1969) pp. 1293–1298
[a6] D.C. Ravenel, "Complex cobordism and stable homotopy groups of spheres" , Pure and Applied Mathematics , 121 , Acad. Press (1986)
[a7] D.C. Ravenel, "Nilpotence and periodicity in stable homotopy theory" , Annals of Math. Stud. , 128 , Princeton Univ. Press (1992)
[a8] W.S. Wilson, "Brown–Peterson homology, an introduction and sampler" , Regional Conf. Ser. Math. , 48 , Amer. Math. Soc. (1982)
How to Cite This Entry:
Brown-Peterson spectrum. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Brown-Peterson_spectrum&oldid=46166
This article was adapted from an original article by S.O. Kochman (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article